Whakaoti i te x+3y=6;5x-2y=13

$$x + 3 y = 6 ; 5 x - 2 y = 13$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$x+3y=6,5x-2y=13$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x+3y=6$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$x=-3y+6$$

Whakakapia te -3y+6 mō te x ki tērā atu whārite, 5x-2y=13.

$$5\left(-3y+6\right)-2y=13$$

Whakareatia 5 ki te -3y+6.

$$-15y+30-2y=13$$

Tāpiri -15y ki te -2y.

$$-17y+30=13$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$-17y=-17$$

Whakawehea ngā taha e rua ki te -17.

$$y=1$$

Whakaurua te 1 mō y ki x=-3y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3+6$$

Tāpiri 6 ki te -3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x+3y=6,5x-2y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&3\\5&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}1&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-3\times 5}&-\frac{3}{-2-3\times 5}\\-\frac{5}{-2-3\times 5}&\frac{1}{-2-3\times 5}\end{matrix}\right)\left(\begin{matrix}6\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}&\frac{3}{17}\\\frac{5}{17}&-\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}6\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 6+\frac{3}{17}\times 13\\\frac{5}{17}\times 6-\frac{1}{17}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$x+3y=6,5x-2y=13$$

Kia ōrite ai a x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5x+5\times 3y=5\times 6,5x-2y=13$$

Whakarūnātia.

$$5x+15y=30,5x-2y=13$$

Me tango 5x-2y=13 mai i 5x+15y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+15y+2y=30-13$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y+2y=30-13$$

Tāpiri 15y ki te 2y.

$$17y=30-13$$

Tāpiri 30 ki te -13.

$$17y=17$$

Whakawehea ngā taha e rua ki te 17.

$$y=1$$

Whakaurua te 1 mō y ki 5x-2y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-2=13$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$5x=15$$

Whakawehea ngā taha e rua ki te 5.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$